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12    UNBOUNDED NORMAL OPERATORS         417

since ||j9 • x\\ g ||*||, it follows from the above and from (15.11.7) that Sp(£)
is contained in [0, 1]. Moreover, the relation (x \ B - x) = 0 implies B • x = 0,
hence * = 0, and therefore the hermitian form (x,y)t-+(B • x\y) is non-
degenerate.

We shall prove next that dom(T*T) is dense in E. If T'is the restriction of
7" to dom(r*T), it will be enough to show that r(T') is dense in F(T), since
dom(T*T) is the first projection of r(T'), and dom(T) is dense in E. To see
that the subspace r(T') of the Hilbert space T(T) is dense in r(T), it is enough
to show that if a vector (u, T - u) e F(T) is orthogonal to r(T'), then it is zero.
Now this condition is

((*, r • i0 1 (*,r •!>)) = <)

for all v E dom(r*T), or equivalently (u \ v) + (T - u \ T- v) = 0; or, since
T • v e dom(T*), (u \ v) + (u \ T* T- v) = 0, that is to say,

But /+ T*T maps dom(T*T) onto E. Hence u = 0, as required.

Finally, since B is self-adjoint, F(B) is the orthogonal supplement of
J(r(B)). Since T(B) is the image of T(/ + T*T) under the symmetry operator
S:(x,y)\-*(y,x)9 and since JS = -SJ, it follows that T(/-f T*T) is the
orthogonal supplement of J(T(I + T*TJ); in other words (15.12.4)
(/ + J*T)* = 7+ r*r, or equivalently (J*r)* = T*T.

(15.12.7)    A not necessarily bounded operator T is said to be normal if it is
closed, if dom(T) is dense in E and and if T*T= 7T* (which, we recall,
implies by definition that dom(T*T) = dom(TT*)). We say that T is self-
adjoint if dom(T) is dense in E and if T* = T (which implies that T is closed
(15.12.4)). Clearly a self-adjoint operator is normal. It follows from (15.12.6)
that, if Tis any closed operator such that dom(T) is dense in E, then T*Tand
TT* are self-adjoint.

The following theorem reduces the problem of the structure of (un-
bounded) normal operators to that of continuous normal operators:

(15.12.8)    Let E be a separable Hilbert space.

(i)   If N is a not necessarily bounded normal operator, then dom(TV) =
dom(N*) and \\N-x\\ = ||tf* • x|| for all xeAom(N). The space E is the
Hilbert sum of a family (En) of closed subspaces, such that En c dom(TV) and
En Ğr stable under N and N * for all n, so that the restriction Nn ofN to En is a
continuous normal operator.

(ii)    Conversely, let En be a sequence of 'closed subspaces ofE9 such that E is
the Hilbert sum of the En . For each n, let Nn be a continuous normal operator onuch that C(E) c: dom(r*). Finally,e restriction of G112